Local and global existence of solutions to the nonlocal Whitham equation on half-line. (Q5958914)

The authors of this interesting paper study the following initial-boundary value problem for the nonlocal Whitham equation NEWLINE\[NEWLINE u_t+uu_x+\mathbf K u=0, \quad (t,x)\in {\mathbb R}^+\times {\mathbb R}^+, NEWLINE\]NEWLINE under the initial condition \(u(x,0)=\bar u(x)\), \(x\in{\mathbb R}^+\), \(\partial_x^{j-1}u(0,t)=\tilde u_j(t)\) for \(j=1,2,...,l\) (\(l\leq n\)). \(\mathbf K\) is a pseudodifferential operator on the half-line, NEWLINE\[NEWLINE {\mathbf K}u =(2\pi i)^{-1}\int\limits_{-i\infty }^{+i\infty }\exp\{px\}K(p)\biggl(\hat u(p,t)-\sum\limits_{j=1}^{[\alpha]}\partial_x^{j-1}u(0,t)p^{-j}\,dp\biggr),NEWLINE\]NEWLINE where \(\hat u(p)\) is the Laplace transformation of \(u\), \(\alpha \) is the order of the operator \(\mathbf K\), \([\alpha]\) denotes the largest integer less than or equal to \(\alpha \). The symbol \(K(p)\) of the operator \(\mathbf K\) is an analytic one-valued function defined in the half-complex plane. The authors formulate the local existence of the solutions of the problem without the condition containing the partial derivatives \(\partial_x^{j-1}u(0,t)=\tilde u_j(t)\). Some sufficient conditions for global existence of the solutions of the considered problem are shown. Asymptotics of the solutions for a large time are given as well.